The lifespans of lions in a particular zoo are normally distributed. The average lion lives $11.6$ years; the standard deviation is $1.5$ years. Use the empirical rule (68-95-99.7%) to estimate the probability of a lion living longer than $14.6$ years.
Solution: $11.6$ $10.1$ $13.1$ $8.6$ $14.6$ $7.1$ $16.1$ $95\%$ $2.5\%$ $2.5\%$ We know the lifespans are normally distributed with an average lifespan of $11.6$ years. We know the standard deviation is $1.5$ years, so one standard deviation below the mean is $10.1$ years and one standard deviation above the mean is $13.1$ years. Two standard deviations below the mean is $8.6$ years and two standard deviations above the mean is $14.6$ years. Three standard deviations below the mean is $7.1$ years and three standard deviations above the mean is $16.1$ years. We are interested in the probability of a lion living longer than $14.6$ years. The empirical rule (or the 68-95-99.7 rule) tells us that $95\%$ of the lions will have lifespans within 2 standard deviations of the average lifespan. The remaining $5\%$ of the lions will have lifespans that fall outside the shaded area. Because the normal distribution is symmetrical, half $({2.5\%})$ will live less than $8.6$ years and the other half $({2.5\%})$ will live longer than $14.6$ years. The probability of a particular lion living longer than $14.6$ years is ${2.5\%}$.